A note on Galois modules and the algebraic fundamental group of projective curves

2002/05/01 by Amílcar Pacheco, Amilcar Pacheco, Pacheco, Amilcar
Computer Science · Mathematics · #14G15 #14H30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:14G15 #msc:14H30

paper · pdf · doi:10.48550/arxiv.math/0205008

8 pages

arxiv created 2002/05/01 · openalex publication_date 2002/05/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a smooth projective connected curve of genus g≥ 2 defined over an algebraically closed field k of characteristic p>0. Let G be a finite group, P a Sylow p-subgroup of G and N_G(P) its normalizer in G. We show that if there exists an étale Galois cover Y→ X with group N_G(P), then G is the Galois group wan étale Galois cover Y\toX, where the genus of X depends on the order of G, the number of Sylow p-subgroups of G and g. Suppose that G is an extension of a group H of order prime to p by a p-group P and X is defined over a finite field \mathbbF_q large enough to contain the |H|-th roots of unity. We show that integral idempotent relations in the group ring ℂ[H] imply similar relations among the corresponding generalized Hasse-Witt invariants.

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